Nuprl Lemma : set-metric-subspace2
∀[X:Type]. ∀[d:metric(X)]. ∀[P,Q:X ⟶ ℙ].
(metric-subspace({x:X| Q[x]} ;d;{x:X| P[x]} )) supposing ((∀x,y:X. (P[x] ⇒ y ≡ x ⇒ P[y])) and (∀x:X. (P[x] ⇒ Q[x])\000C))
Proof
Definitions occuring in Statement :
metric-subspace: metric-subspace(X;d;A),
meq: x ≡ y,
metric: metric(X),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
metric-subspace: metric-subspace(X;d;A),
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T}
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[P,Q:X {}\mrightarrow{} \mBbbP{}].
(metric-subspace(\{x:X| Q[x]\} ;d;\{x:X| P[x]\} )) supposing ((\mforall{}x,y:X. (P[x] {}\mRightarrow{} y \mequiv{} x {}\mRightarrow{} P[y])) and (\000C\mforall{}x:X. (P[x] {}\mRightarrow{} Q[x])))
Date html generated:
2020_05_20-AM-11_42_51
Last ObjectModification:
2019_11_22-AM-10_22_23
Theory : reals
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